Math 321 - Real Variables II - Spring 2015
Instructor: Malabika Pramanik
Office: 214 Mathematics Building
E-mail: malabika at math dot ubc dot ca
Lectures: Mon,Wed,Fri 9:00 AM to 10:00 AM in Room 104 Mathematics Building.
Office hours: Mon 10-11, Wed 11-12 or by appointment.
- Homework 1 , due on Friday January 16 at the beginning of lecture.
- Homework 2 , due on Friday January 23 at the beginning of lecture.
- Homework 3 , due on Friday January 30 at the beginning of lecture.
Here is a list of exercises from the textbook based on the weekly lecture material. They are not meant to be turned in for grading, but it is recommended that you work through them to get a better understanding of the topics covered. Some midterm and final exam problems may be modelled on these exercises.
- Week 1 : Chapter 7, Exercises 1, 2, 3, 5, 9, 13, 18.
Week-by-week course outline
Here is a rough guideline of the course structure, arranged by week. The textbook pages are mentioned as a reference and as a reading guide. The treatment of these topics in lecture may vary somewhat from that of the text.
- Week 1 (pages 143-154 of the textbook):
- Pointwise and uniform convergence
- Definitions and examples
- Interchanging limits.
- Week 2 (pages 143-154 of the textbook):
- Applications of pointwise and uniform convergence:
- Weierstrass M-test
- A continuous but nowhere differentiable function
- A space-filling curve
- Separability of the space of continuous functions on [0,1]
- Week 3 (pages 143-154 of the textbook):
- Density of polygonal functions in the space C[0,1]
- A quick review of uniform continuity
- Bernstein's proof of the Weierstrass approximation theorem
- Week 4 (pages 154-165, 188-190 of the textbook):
- Approximation of continuous periodic functions by trigonometric polynomials
- Arzela-Ascoli theorem
- Week 5 (pages 154-165 of the textbook):
- Arzela-Ascoli theorem
- Algebras and lattices
- The Stone-Weierstrass theorem
- Week 6 (see the textbooks in real analysis by T. Hildebrandt, or N. Carothers) :
- Towards Riemann-Stieltjes integral
- Functions of bounded variation
- Jordan's theorem
- Week 7 :
- Week 8 :
- Week 9 (pages 120-133 of the textbook):
- The Riemann-Stieltjes integral
- Riemann's condition for Riemann-Stieltjes integrability
- An integral formula for total variation
- Week 10 (pages 120-133 of the textbook) :
- The space of Riemann-Stieltjes integrable functions
- Closure under uniform convergence
- General integrators
- Week 11 (pages 120-133 of the textbook) :
- Integration by parts
- Integrators of bounded variation
- Riesz representation theorem
- Week 12 (pages 185-192 of the textbook):
- Fourier series
- L^2 convergence of Fourier series
- Bessel's inequality
- Plancherel's theorem
- Week 13 (pages 204-223 of the textbook):
- Linear transformations and differentiation
- The inverse function theorem
- The implicit function theorem